A TOURNAMENT VERSION OF THE LOVÁSZ PATH REMOVAL
CONJECTURE
JAEHOON KIM, DANIELA KÜHN, AND DERYK OSTHUS
Abstract. We prove the following tournament version of the Lovász path removal conjecture:
for each k ∈ N there exists g(k) ∈ N so that every strongly g(k)-connected tournament T has
the property that for any ordered pair x, y of vertices there is a path P from x to y in T for
which T \ V (P ) is strongly k-connected. In fact, we prove the following stronger statement,
which provides a tournament analogue of a long-standing conjecture of Thomassen: suppose
that T is a strongly 107 k6 m-connected tournament. Then for every set M of m vertices in T ,
there is a partition V1 , V2 of V (T ) such that (i) M ⊆ V1 , (ii) for i = 1, 2 the subtournament
T [Vi ] is strongly k-connected, and (iii) every vertex in V1 has at least k out-neighbours and at
least k in-neighbours in V2 .
1. Introduction
The famous Lovász path removal conjecture states that for every k ∈ N there exists g(k) ∈ N
such that for every pair x, y of vertices in a g(k)-connected graph G we can find an induced
path P joining x and y in G for which G \ V (P ) is k-connected. It is not hard to show that
g(1) = 3. Chen, Gould and Yu [1] as well as Kriesell [4] independently showed that g(2) = 5.
In general, the conjecture is still wide open (a version for edge-connectivity was proved in [3]).
More generally, one can also ask for the existence of a non-separating subdivision of a graph H
with prescribed branch vertices such that the paths joining the branch vertices are induced (the
path removal conjecture then corresponds to the special case when H consists of a single edge).
We prove such a result for tournaments. The natural tournament analogue of an induced
path is a backwards transitive path: here a directed path P = x1 . . . xt in a tournament T is
backwards-transitive if xi xj is an edge of T whenever i ≥ j + 2.
Theorem 1.1. Let k, m ∈ N. Suppose that T is a strongly 1023 k 6 m13 -connected tournament,
that M is a set of m vertices in T , that H is a digraph on m vertices and that φ is a bijection
from V (H) to M . Then T contains a subdivision H ∗ of H such that
(i) for each h ∈ V (H) the branch vertex of H ∗ corresponding to h is φ(h),
(ii) T \ V (H ∗ ) is k-connected,
(iii) for every edge e of H, the path Pe of H ∗ corresponding to e is backwards-transitive.
Our proof shows that in the case when H is an edge it suffices for T to be strongly 2 · 107 k 6 connected. We will derive Theorem 1.1 from the following result on tournament partitions:
Theorem 1.2. Let T be a tournament and k, m ∈ N. If T is strongly 107 k 6 m-connected then for
any set M ⊆ V (T ) with |M | = m, there exists a partition V1 , V2 of V (T ) such that M ⊆ V1 , T [V1 ]
Date: November 6, 2014.
The research leading to these results was partially supported by the European Research Council under the
European Union’s Seventh Framework Programme (FP/2007–2013) / ERC Grant Agreements no. and 306349
(J. Kim and D. Osthus) as well as 258345 (D. Kühn).
1
2
JAEHOON KIM, DANIELA KÜHN, AND DERYK OSTHUS
and T [V2 ] are both strongly k-connected, and every vertex in V1 has at least k out-neighbours
and at least k in-neighbours in V2 .
We have made no attempt to optimize the bound on the connectivity in Theorem 1.2. (It
would be straightforward to obtain minor improvements at the expense of more careful calculations.) On the other hand, it would be interesting to obtain the correct order of magnitude (in
terms of m and k) for the connectivity bound.
Kühn, Osthus and Townsend [7] earlier proved the weaker result that every strongly 108 k 6 log(4k)connected tournament T has a vertex partition V1 , V2 such that T [V1 ] and T [V2 ] are both
strongly k-connected (with some control over the sizes of V1 and V2 ). This proved a conjecture
of Thomassen. [7] raised the question whether this can be extended to digraphs. A graph version
of this was already proved much earlier by Hajnal [2] and Thomassen [10].
As described later, our proof of Theorem 1.2 develops ideas in [7]. These in turn are based
on the concept of robust linkage structures which were introduced in [5] to prove a conjecture of
Thomassen on edge-disjoint Hamilton cycles in highly connected tournaments. Further (asymptotically optimal) results leading on from these approaches were obtained by Pokrovskiy [8, 9].
Theorem 1.2 is a tournament analogue of the following long-standing conjecture of Thomassen
[11].
Conjecture 1.3. For every k ∈ N there exists f (k) ∈ N such that if G is a f (k)-connected
graph and M ⊆ V (G) consists of k vertices then there exists a partition V1 , V2 of V (G) such
that M ⊆ V1 , both G[V1 ] and G[V2 ] are k-connected, and each vertex in V1 has at least k
neighbours in V2 .
The case |M | = 2 would already imply the path removal conjecture. The case M = ∅ was
proved in [6]. It implies the existence of non-separating subdivisions (without prescribed branch
vertices) in highly connected graphs.
2. Notation and tools
Given k ∈ N, we let [k] := {1, . . . , k} and log k := log2 k. We write V (G) and E(G) for the
set of vertices and the set of edges in a digraph G. We let |G| := |V (G)|. If u, v ∈ V (G) we
+
write uv for the directed edge from u to v. We write d−
G (v) and dG (v) for the in-degree and the
out-degree of a vertex v in G. We write δ − (G) and δ + (G) for the minimum in-degree and the
minimum out-degree of G and let δ 0 (G) := min{δ − (G), δ + (G)}. A set A ⊆ V (G) in-dominates
a set B ⊆ V (G) if for every vertex b ∈ B there exists a vertex a ∈ A such that ba ∈ E(G).
Similarly, we say that A out-dominates B if for every vertex b ∈ B there exists a vertex a ∈ A
such that ab ∈ E(G). We say that a tournament T is transitive if we may enumerate its vertices
v1 , . . . , vm such that vi vj ∈ E(T ) if and only if i < j. In this case we call v1 the source of T and
vm the sink of T . When referring to subpaths of tournaments, we always mean that these paths
are directed (i.e. consistently oriented). The length of a path is the number of its edges. We say
that two paths are disjoint if they are vertex-disjoint. A tournament T is strongly k-connected
if |T | > k and for every set F ⊆ V (T ) with |F | < k and every ordered pair x, y of vertices
in V (T ) \ F there exists a path from x to y in T − F . A tournament T is called k-linked if
|T | ≥ 2k and whenever x1 , . . . , xk , y1 , . . . , yk are 2k distinct vertices of T there exist disjoint
paths P1 , . . . , Pk such that Pi is a directed path from xi to yi for each i ∈ [k].
We now collect the tools which we need in our proof of Theorem 1.2. We will use the following
well known fact.
A TOURNAMENT VERSION OF THE LOVÁSZ PATH REMOVAL CONJECTURE
3
Proposition 2.1. Let k ∈ N and let T be a tournament. Then T contains less than 2k vertices
of out-degree less than k, and T contains less than 2k vertices of in-degree less than k.
The following proposition is a straightforward consequence of the definition of linkedness.
Proposition 2.2. Let k ∈ N. Then a tournament T is k-linked if and only if |T | ≥ 2k and
whenever (x1 , y1 ), . . . , (xk , yk ) are ordered pairs of (not necessarily distinct) vertices of T , there
exist distinct internally disjoint paths P1 , . . . , Pk such that for all i ∈ [k] we have that Pi is a
directed path from xi to yi and that {x1 , . . . , xk , y1 , . . . , yk } ∩ V (Pi ) = {xi , yi }.
We will also use the following bound from [5] on the connectivity which forces a tournament
to be highly linked.
Theorem 2.3. For each k ∈ N every strongly 452k-connected tournament is k-linked.
The following two lemmas from [7] guarantee that every tournament contains almost outdominating and almost in-dominating sets which are not too large.
Lemma 2.4. Let T be a tournament, let v ∈ V (T ) and c ∈ N. Then there exist disjoint sets
A, E ⊆ V (T ) such that the following properties hold:
(i) 1 ≤ |A| ≤ c and T [A] is a transitive tournament with sink v,
(ii) A out-dominates V (T ) \ (A ∪ E),
(iii) |E| ≤ (1/2)c−1 d−
T (v).
The next lemma follows immediately from Lemma 2.4 by reversing the orientations of all
edges.
Lemma 2.5. Let T be a tournament, let v ∈ V (T ) and c ∈ N. Then there exist disjoint sets
B, E ⊆ V (T ) such that the following properties hold:
(i) 1 ≤ |B| ≤ c and T [B] is a transitive tournament with source v,
(ii) B in-dominates V (T ) \ (B ∪ E),
(iii) |E| ≤ (1/2)c−1 d+
T (v).
We will also need the following observation, which guarantees a small set Z of vertices in a
tournament such that every vertex outside Z has many out- and in-neighbours in Z.
Proposition 2.6. Let k, n ∈ N and let T be a tournament on n ≥ 16 vertices. Then there
is a set Z ⊆ V (T ) of size |Z| ≤ 3k log n such that each vertex in V (T ) \ Z has at least k
out-neighbours and at least k in-neighbours in Z.
Proof.
We may assume that n ≥ 3k log n. Let c := dlog ne + 1 ≤ (3 log n)/2. Note that
Lemma 2.5 implies that T contains an in-dominating set V1 of size at most c. Apply Lemma 2.5
again to T \ V1 to find an in-dominating set V2 of T \ V1 with size at most c. Continue in this
way to obtain disjoint sets V1 , . . . , Vk . Now apply Lemma 2.4 repeatedly to obtain disjoint sets
U1 , . . . , Uk , each of size at most c, such that each Ui is an out-dominating set in T \(U1 ∪· · ·∪Ui−1 ).
We can take Z := V1 ∪ · · · ∪ Vk ∪ U1 · · · ∪ Uk .
Recall that a subpath Q = q1 . . . q|Q| of a tournament T is backwards-transitive if qi qj ∈ E(T )
whenever i ≥ j + 2. The following lemma is a slight strengthening of Lemma 2.7 in [7]. The
proof is identical to that in [7], so we omit it here.
4
JAEHOON KIM, DANIELA KÜHN, AND DERYK OSTHUS
Lemma 2.7. Let k, ` ∈ N, let T be a tournament and let Q1 , . . . , Q` be disjoint backwardstransitive paths in T such that |Qj | ≥ k + 1 for all j ∈ [`] and V (T ) = V (Q1 ∪ · · · ∪ Q` ). Let
U 0 be the set consisting of the first k + 1 vertices in Qj for all j ∈ [`] and let W 0 be the set
consisting of the last k + 1 vertices in Qj for all j ∈ [`]. Then there exist sets U, W satisfying
the following properties:
• U ⊆ U 0 ⊆ V (T ) and W ⊆ W 0 ⊆ V (T ),
• |U |, |W | ≤ 2k(k + 1),
• for any set F ⊆ V (T ) of size at most k − 1, and for every vertex v in V (T ) \ F , there
exists a directed path (possibly of length 0) in T [(U 0 ∪ {v}) \ F ] from v to a vertex in U
and a directed path in T [(W 0 ∪ {v}) \ F ] from a vertex in W to v.
Note that U 0 and W 0 may not be disjoint, and |U 0 | = |W 0 | = `(k + 1).
3. Proofs of Theorems 1.1 and 1.2
Before we prove Theorem 1.2, we will show how it can be used to derive Theorem 1.1.
Proof of Theorem 1.1. Apply Theorem 1.2 to obtain a partition V1 , V2 of V (T ) such that
M ⊆ V1 , T [V1 ] and T [V2 ] are both strongly 452km2 -connected, and every vertex in V1 has at
least k out-neighbours and at least k in-neighbours in V2 . Theorem 2.3 now implies that T [V1 ] is
m2 -linked. Together with Proposition 2.2 this in turn implies that T [V1 ] contains a subdivision
H ∗ of H such that for each h ∈ V (H) the branch vertex of H ∗ corresponding to h is φ(h).
By shortening the paths between the branch vertices if necessary, we may assume that they
are backwards-transitive. Since every vertex in V1 has at least k out-neighbours and at least k
in-neighbours in V2 it follows that T [V2 ∪ (V1 \ V (H ∗ ))] is strongly k-connected, as desired.
We now give a brief idea of the argument in the proof of Theorem 1.2 under the much stronger
assumptions that k log n and |M | = 1. In this case we can find 2k disjoint sets A1 , . . . , A2k ⊆
V (T ) of size o(k) which are out-dominating. We can also find 2k sets B1 , . . . , B2k ⊆ V (T ) of
size o(k) which are in-dominating such that all the Bi are disjoint from each other and from
A1 , . . . , A2k . Moreover, we can choose these sets in such a way that each Ai and each Bi induces
a transitive subtournament of T . We now use the fact that T is (107 k 6 /452)-linked to find, for
each i ∈ [2k], a path Pi from the sink of Bi to the source of Ai such that all the Pi are pairwise
disjoint. We now assign Ai ∪ Bi ∪ V (Pi ) to V1 for all i ≤ k and to V2 for all i > k. We assign
the remaining vertices arbitrarily. By relabeling V1 and V2 if necessary, we may assume that
M ⊆ V1 .
It is easy to see that both T [V1 ] and T [V2 ] are strongly k-connected. Indeed, consider some
F ⊆ V1 with |F | < k. So there exists i ∈ [k] such that F avoids Ai ∪ Bi ∪ V (Pi ). Consider any
x, y ∈ V1 \ F . Since Bi is in-dominating, there is an edge from x to some x0 ∈ Bi . Similarly,
since Ai is out-dominating, there is an edge from some y 0 ∈ Ai to y. Then Pi , xx0 , y 0 y together
with the edge from x0 to the sink of Bi and the edge from the source of Ai to y 0 form a path
in T [V1 \ F ] from x to y, as required. A similar argument shows that T [V2 ] is k-connected too.
Moreover, each x ∈ V1 has k in-neighbours and k out-neighbours in V2 since x receives an edge
from Ai and sends an edge to Bi for all i > k.
In general, the problem with this approach is that we cannot guarantee such (small) dominating sets when k is bounded. However, we can still find small sets which dominate a large
proportion of V (T ). With some new ideas one can use these to ensure strong k-connectivity of
both T [V1 ] and T [V2 ] as well as high in- and outdegree of the vertices in V1 from and to V2 .
Significant additional difficulties arise when |M | > 1.
A TOURNAMENT VERSION OF THE LOVÁSZ PATH REMOVAL CONJECTURE
5
Proof of Theorem 1.2. Let X := {x1 , x2 , . . . , x20k } ⊆ V (T ) \ M consist of 20k vertices whose
in-degree in T is as small as possible, and let Y := {y1 , y2 , . . . , y20k } be a set of 20k vertices in
V (T ) \ (M ∪ X) whose out-degree in T is as small as possible. Define
δ̂ − (T ) :=
min
v∈V (T )\(M ∪X)
d−
T (v)
and
δ̂ + (T ) :=
min
v∈V (T )\(M ∪Y )
d+
T (v).
Let c := dlog (80k)e+1 ≤ 9k. Apply Lemmas 2.4 and 2.5 with parameter c repeatedly (removing
M and the dominating sets each time) to obtain disjoint sets of vertices A1 , . . . , A20k , B1 , . . . , B20k
and sets of vertices EA1 , . . . , EA20k , EB1 , . . . , EB20k satisfying the following properties for all
S20k
S19k
S
i ∈ [20k], where we write D := 20k
i=19k+1 (Ai ∪Bi ):
i=1 (Ai ∪Bi ) and D2 :=
i=1 (Ai ∪Bi ), D1 :=
(D1) 1 ≤ |Ai | ≤ c and T [Ai ] is a transitive tournament with sink xi ,
(D2) 1 ≤ |Bi | ≤ c and T [Bi ] is a transitive tournament with source yi ,
(D3) Ai out-dominates V (T ) \ (M ∪ D ∪ EAi ) in T ,
(D4) Bi in-dominates V (T ) \ (M ∪ D ∪ EBi ) in T ,
(D5) |EAi | ≤ (1/2)c−1 δ̂ − (T ),
(D6) |EBi | ≤ (1/2)c−1 δ̂ + (T ).
0 := E
0
Let EA := EA1 ∪ · · · ∪ EA20k , EB := EB1 ∪ · · · ∪ EB20k , EA
A19k+1 ∪ · · · ∪ EA20k , EB :=
0
0
0
EB19k+1 ∪ · · · ∪ EB20k , E := EA ∪ EB and E := EA ∪ EB . Note that
c−1
1
δ̂ − (T )
δ̂ + (T )
0
0
(3.1)
|EA | ≤ |EA | ≤ 20k
δ̂ − (T ) ≤
and |EB
| ≤ |EB | ≤
2
4
4
by our choice of c. Moreover, we may assume that |EA | ≤ |EB |. (The case |EA | > |EB | follows
by a symmetric argument.) In particular, this implies that
δ̂ + (T )
.
2
Our aim is to use the almost-dominating sets Ai , Bi in order to construct the desired partition
V1 , V2 of V (T ). More precisely, we will iteratively colour the vertices of T with colours α and
β, and at each step Vα will consist of all vertices of colour α and Vβ is defined similarly. At
the end of our argument, every vertex of T will be coloured either with α or with β, i.e. Vα , Vβ
will form a partition of V (T ). Our aim is to colour the vertices in such a way that we can take
V1 := Vα and V2 := Vβ . We start with no vertices of T coloured, and we then colour the vertices
S
S20k
in M ∪ D1 = M ∪ 19k
i=19k+1 (Ai ∪ Bi ) by β.
i=1 (Ai ∪ Bi ) by α and the vertices in D2 =
At each step and for each γ ∈ {α, β}, we call a vertex v ∈ Vγ forwards-safe if for any set
F 63 v of at most k − 1 vertices, there is a directed path (possibly of length 0) in T [Vγ \ F ]
from v to Vγ \ (M ∪ D ∪ EB ∪ F ). Similarly, we say that v ∈ Vγ is backwards-safe if for any set
F 63 v of at most k − 1 vertices, there is a directed path (possibly of length 0) in T [Vγ \ F ] from
Vγ \ (M ∪ D ∪ EA ∪ F ) to v. We will call a vertex v ∈ Vγ partition-safe if either v ∈
/ M ∪D∪E
or γ = β or v has at least k out-neighbours and k in-neighbours of colour β. Finally, we call
a vertex safe if it is forwards-safe, backwards-safe and partition-safe. Note that the following
properties are satisfied at every step (for each γ ∈ {α, β}):
(S1) all coloured vertices in V (T ) \ (M ∪ D ∪ E) are safe,
(S2) all coloured vertices in V (T ) \ (M ∪ D ∪ EB ) are forwards-safe and all coloured vertices
in V (T ) \ (M ∪ D ∪ EA ) are backwards-safe,
(S3) if v ∈ Vγ has at least k forwards-safe out-neighbours of colour γ then v itself is forwardssafe; the analogue holds if v has at least k backwards-safe in-neighbours of colour γ,
(3.2)
|E 0 | ≤ |E| ≤ |EA | + |EB | ≤ 2|EB | ≤
6
JAEHOON KIM, DANIELA KÜHN, AND DERYK OSTHUS
0 ) which have at least k out-neighbours in V
(S4) all coloured vertices in Vα \ (M ∪ D ∪ EA
β
are partition-safe,
(S5) if v ∈ Vγ is safe and in the next step we enlarge Vγ by colouring some more (previously
uncoloured) vertices then v is still safe.
0 ) has
Indeed, to check (S4) note that (D3) implies that every vertex in V (T ) \ (M ∪ D ∪ EA
an in-neighbour in As for each 19k < s ≤ 20k and that all vertices in these As are coloured β.
In what follows, by a (partial) colouring of the vertices of T we always mean a colouring with
colours α and β in which all the vertices in M ∪ D1 are coloured α and all the vertices in D2
are coloured β.
Claim 1: Suppose that there are distinct indices i1 , . . . , ik ∈ [19k], distinct indices i01 , . . . , i0k ∈
[19k] and subpaths Q1 , . . . , Qk and P19k+1 , . . . , P20k of T satisfying the following properties:
• for each s ∈ [k] the path Qs joins the sink of Bi0s to the source of Ais ,
• for each 19k < s ≤ 20k the path Ps joins the sink of Bs to the source of As ,
• the paths Q1 , . . . , Qk and P19k+1 , . . . , P20k are disjoint from each other and meet D ∪ M
only in their endvertices.
Suppose that we have coloured all vertices of T such that
• every vertex in M ∪ D1 ∪ V (Q1 ) ∪ · · · ∪ V (Qk ) is coloured α,
• every vertex in D2 ∪ V (P19k+1 ) ∪ · · · ∪ V (P20k ) is coloured β,
• every vertex is safe.
Then the sets V1 := Vα and V2 := Vβ form a partition of V (T ) as required in Theorem 1.2.
To prove Claim 1, we first show that T [Vα ] is strongly k-connected. So consider any set F
of at most k − 1 vertices and any two vertices x, y ∈ Vα \ F . We need to check that T [Vα \ F ]
contains a path from x to y. Since x is forwards-safe there exists a path Px in T [Vα \ F ] from x
to some vertex x0 ∈ Vα \ (M ∪ D ∪ EB ∪ F ). Similarly, since y is backwards-safe there exists a
path Py in T [Vα \ F ] from some vertex y 0 ∈ Vα \ (M ∪ D ∪ EA ∪ F ) to y. Let s ∈ [k] be such that
F avoids Ais ∪ V (Qs ) ∪ Bi0s . Since x0 ∈
/ M ∪ D ∪ EB , (D4) implies that x0 sends an edge to Bi0s .
0
Similarly, since y ∈
/ M ∪ D ∪ EA , (D3) implies that y 0 receives an edge from Ais . Altogether
this implies that T [V (Px ) ∪ V (Py ) ∪ Ais ∪ V (Qs ) ∪ Bi0s ] ⊆ T [Vα \ F ] contains path from x to y,
as desired.
A similar argument shows that Vβ is strongly k-connected too. It remains to show that any
vertex x ∈ Vα has k in-neighbours and k out-neighbours in Vβ . Since x is partition-safe this is
clear if x ∈ M ∪ D ∪ E. If x ∈
/ M ∪ D ∪ E then (D3) and (D4) together imply that, for every
19k < s ≤ 20k, x sends an edge to Bs ⊆ Vβ and receives an edge from As ⊆ Vβ . This completes
the proof of Claim 1.
Claim 2: Consider a partial colouring of V (T ) and let C denote the set of previously coloured
vertices. (So M ∪ D ⊆ C.) Let Z ⊆ V (T ) \ (M ∪ X ∪ Y ) and N ⊆ V (T ) \ Z and suppose that
9k 2 |Z| + |C ∪ N | ≤ 5 · 106 k 6 m. Then for every colouring of the vertices in Z \ C there is a set
Z 0 ⊆ V (T ) \ (Z ∪ N ∪ C) and a colouring of the vertices in Z 0 such that every vertex in Z ∪ Z 0
is safe and |Z ∪ Z 0 | ≤ 9k 2 |Z|.
To prove Claim 2, note that the strong 107 k 6 m-connectivity of T implies that δ 0 (T ) ≥ 107 k 6 m.
Hence
(3.1)
(3.3)
δ̂ − (T ) − |EA | ≥
δ 0 (T )
δ̂ − (T )
≥
≥ 5 · 106 k 6 m,
2
2
A TOURNAMENT VERSION OF THE LOVÁSZ PATH REMOVAL CONJECTURE
7
and similarly
δ̂ + (T )
≥ 5 · 106 k 6 m.
2
Consider any colouring of Z \ C. Let Zα be the vertices in Z coloured with α and define
Zβ similarly. For each vertex z ∈ Zβ in turn we greedily choose k uncoloured in-neighbours
outside N ∪ EA , and colour them β. Then for each vertex z ∈ Zα in turn we greedily choose 2k
uncoloured in-neighbours outside N ∪ EA , and colour k of them α and k of them β. (We do not
modify C in this process.) To see that we can choose all these vertices to be distinct from each
other, note that the total number of vertices we wish to choose is 2k|Zα | + k|Zβ | ≤ 2k|Z| and
(3.2)
δ̂ + (T ) − |E| ≥
(3.4)
(3.3)
|C ∪ N ∪ Z| + 2k|Z| ≤ 5 · 106 k 6 m ≤ δ̂ − (T ) − |EA |.
For each vertex outside C \Z of colour β in turn we greedily choose k uncoloured out-neighbours
outside N ∪ E, and colour them by β. Now for each vertex outside C \ Z of colour α in turn we
greedily choose 2k uncoloured out-neighbours not in N ∪ E and colour k of them by α and k of
them by β. To see that we can choose such vertices to be distinct from each other, note that
the total number of vertices we wish to choose is at most 2k(1 + 2k)|Z| and
(3.4)
|C ∪ N ∪ Z| + 2k|Z| + 2k(1 + 2k)|Z| ≤ |C ∪ N | + 9k 2 |Z| ≤ 5 · 106 k 6 m ≤ δ̂ − (T ) − |E|.
Let Z 0 be the set of vertices outside C ∪ Z that we coloured. Then Z 0 ∩ N = ∅. Moreover, using
(S1)–(S4) it is easy to check that every vertex in Z ∪ Z 0 is safe. This completes the proof of
Claim 2.
= α, = β
6∈ N ∪ E
Zα 3
6∈ N ∪ E
6∈ N ∪ EA
6∈ N ∪ E
6∈ N ∪ EA
6∈ N ∪ E
Zβ 3
6∈ N ∪ EA
6∈ N ∪ E
6∈ N ∪ E
6∈ N ∪ E
Figure 1 :
The vertices chosen in the case when k = 1 in order to make
one vertex in Zα safe (left) and one vertex in Zβ safe (right).
Recall that we have already coloured all the vertices in M ∪ D1 by α and all the vertices in
D2 by β. Step by step, we will now colour further vertices of T . Our final aim is to arrive at
a colouring of V (T ) which is as described in Claim 1. The first step is to colour some more
vertices in order to achieve that all the coloured vertices are safe. In what follows, when saying
that we colour some additional vertices we always mean that these vertices are uncoloured so
far.
Claim 3: We can colour some additional vertices of T in such a way that every coloured vertex
is safe and the set C1 consisting of all vertices coloured so far satisfies |C1 | ≤ 5000k 4 m.
To prove Claim 3, for every v ∈ {x1 , . . . , x19k , y1 , . . . , y19k } ∪ M in turn we greedily choose
2k uncoloured in-neighbours and 2k uncoloured out-neighbours, all distinct from each other,
8
JAEHOON KIM, DANIELA KÜHN, AND DERYK OSTHUS
and colour k out-neighbours and k in-neighbours by α and the other k out- and in-neighbours
by β. Similarly, for every v ∈ {x19k+1 , . . . , x20k , y19k+1 , . . . , y20k } in turn we greedily choose
k uncoloured in-neighbours and k uncolored out-neighbours, all distinct from each other and
colour them β. Let Z ∗ denote the set of 4k(38k + m) + 4k 2 ≤ 160k 2 m new vertices we just
coloured and let Z := Z ∗ ∪ (D \ (X ∪ Y )). Then |Z| ≤ |Z ∗ | + |D| ≤ 160k 2 m + c · 40k ≤ 520k 2 m.
Apply Claim 2 with N := ∅ to find a set Z 0 of uncoloured vertices and a colouring of these
vertices such that all the vertices in Z ∪ Z 0 are safe and |Z ∪ Z 0 | ≤ 9k 2 · |Z| ≤ 5000k 4 m. Our
choice of Z ∗ and (S3) together now imply that the vertices in X ∪ Y ∪ M are safe as well. This
completes the proof of Claim 3.
Claim 4: There are distinct indices i1 , . . . , ik ∈ [19k], distinct indices i01 , . . . , i0k ∈ [19k] and
subpaths Q1 , . . . , Qk and P19k+1 , . . . , P20k of T satisfying the following properties:
(i) for each s ∈ [k] the path Qs joins the sink of Bi0s to the source of Ais ,
(ii) for each 19k < s ≤ 20k the path Ps joins the sink of Bs to the source of As ,
(iii) the paths Q1 , . . . , Qk and P19k+1 , . . . , P20k are disjoint from each other and meet C1 ⊇
D ∪ M only in their endvertices,
(iv) we can colour the internal vertices of Q1 , . . . , Qk by α, the internal vertices of P19k+1 , . . . , P20k
by β and can colour some additional vertices such that the set C4 of all coloured vertices
satisfies the following properties:
(α) all vertices in C4 are safe,
(β) there is a set Cα ⊆ C4 such that every vertex in Cα is coloured α and the number of
vertices of colour α outside Cα is at most 106 k 6 m,
(γ) every vertex outside C4 which has an in-neighbour in Cα has at least k in-neighbours
coloured β, and every vertex outside C4 which has an out-neighbour in Cα has at least
k out-neighbours coloured β.
We will prove Claim 4 via a sequence of subclaims. For i ∈ [20k] we define an i-path to be
a directed path from the sink of Bi to the source of Ai whose interior vertices lie outside C1 .
Ideally, we would like to find disjoint i-paths Pi (one for each i ∈ [20k]) such that the following
properties hold:
(a) for 19k < i ≤ 20k all interior vertices of Pi can be coloured β,
(b) there are at least k indices i with i ∈ [19k] such that all interior vertices of Pi can be
coloured α,
(c) by colouring some additional vertices we can achieve that all coloured vertices are safe.
However, we are not able to satisfy (b) (and (c)) directly. So instead, for each of the paths Qs
in Claim 4, there will be three paths Pi1 , Pi2 and Pi3 with i1 , i2 , i3 ∈ [19k] such that each Pij is
an ij -path and Qs consists of an initial segment of Pi1 , a middle segment of Pi2 , a final segment
of Pi3 as well as two edges joining these three segments.
More precisely, our strategy is to proceed as follows. For each i ∈ [20k] we will first try to
find a short i-path Pi such that all these i-paths are disjoint. We will then colour the vertices on
these short i-paths as well as some additional vertices such that (a)–(c) are satisfied for the set
Ishort of all indices i for which we have been able to choose a short i-path (see Claim 4.1). This
provides some of the paths required in Claim 4. To find the remaining paths, for all i ∈
/ Ishort we
will choose 1000k 4 i-paths Qi,1 , . . . , Qi,1000k4 such that all these paths are internally disjoint from
each other. We will then show that for each i ∈
/ Ishort with i > 19k we can take the Pi required in
Claim 4 to be some Qi,j , whereas each remaining path Qs still required in Claim 4 will consist of
one segment from each of three different paths Qi1 ,j1 , Qi2 ,j2 , Qi3 ,j3 with i1 , i2 , i3 ∈ [19k]\Ishort , as
A TOURNAMENT VERSION OF THE LOVÁSZ PATH REMOVAL CONJECTURE
9
described before. The reason why we start with 19k indices to choose the k paths Qs in Claim 4
and why we choose many i-paths for each i ∈
/ Ishort is that we need some extra flexibility in
order to be able to satisfy part (vi) of Claim 4.
We will now choose the short i-paths. So let Pshort be a collection of paths consisting of at
most one i-path for each i ∈ [20k] such that all these paths are disjoint from each other, each
path has length at most 6k + 10 and, subject to this, |Pshort | is as large as possible. Let Ishort be
the set of all those indices i ∈ [20k] for which Pshort contains an i-path, let Ishort,α := Ishort ∩[19k]
and Ishort,β := Ishort \ Ishort,α . Moreover, set Ilong := [20k] \ Ishort , Ilong,α := Ilong ∩ [19k] and
Ilong,β := Ilong \ Ilong,α . For each i ∈ Ishort let Pi denote the i-path contained in Pshort . We will
call all these i-paths short. Let Vshort be the set of all internal vertices of Pi for all i ∈ Ishort .
Recall that the definition of an i-path implies that all the vertices in Vshort are uncoloured so
far (i.e. Vshort ∩ C1 = ∅).
Claim 4.1: We may colour all vertices in Vshort as well as some additional vertices of T such
that the following properties hold:
(i) for each i ∈ Ishort,α all the vertices on Pi are coloured α,
(ii) for each i ∈ Ishort,β all the vertices on Pi are coloured β,
(iii) the set C2 consisting of all vertices coloured so far has size |C2 | ≤ 8000k 4 m and all vertices
in C2 are safe.
Note that |Vshort | ≤ 20k(6k + 9) ≤ 300k 2 . Together with Claim 2 (applied with N := ∅ and
Z := Vshort ) and Claim 3 this implies Claim 4.1.
Claim 4.2: We may assume that |Ishort,α | < k, and hence |Ilong,α | > 18k.
To prove Claim 4.2, suppose that |Ishort,α | ≥ k. Colour all uncoloured vertices by β. Then
|Vα | ≤ 8000k 4 m by Claim 4.1(iii). Since T is strongly 107 k 6 m-connected and 107 k 6 m − |Vα | ≥
107 k 6 m − 8000k 4 m > k, it follows that T [Vβ ] is strongly k-connected and that every vertex in
Vα has at least k in-neighbours and k out-neighbours in Vβ . Using the facts that T [Vα ] contains
D1 as well as disjoint i-paths for all i ∈ Ishort,α and that all the vertices in Vα are safe, a
similar argument as in the proof of Claim 1 shows that T [Vα ] is strongly k-connected too. So
the partition Vα , Vβ is as desired in Theorem 1.2. This completes the proof of Claim 4.2.
Recall from Claim 4.1(iii) that the set C2 of coloured vertices has size at most 8000k 4 m.
So all uncoloured vertices together with the sinks of the Bi and the sources of the Ai for all
i ∈ Ilong induce a strongly (904 · 104 k 5 )-connected subtournament T 0 of T (with some room
to spare). Theorem 2.3 implies that T 0 is 2 · 104 k 5 -linked. Together with Proposition 2.2 this
implies that for each i ∈ Ilong we can find 1000k 4 i-paths in T 0 such that all these 1000k 4 |Ilong |
paths are internally disjoint and the internal vertices on all these paths lie outside C2 . We
choose such a collection of paths which minimizes the size of the set Vlong consisting of all the
internal vertices on these paths. Let Qi,j denote the jth i-path we chose (for all i ∈ Ilong and
all j ∈ [1000k 4 ]). Note that each Qi,j must have length at least 6k + 11 since i ∈ Ilong . Write
|Q
|
|Q
|
0 q 1 . . . q i,j . So q 0 is the the sink of B and q i,j is the source of A . Observe that
Qi,j = qi,j
i
i
i,j
i,j
i,j
i,j
the minimality of |Vlong | implies the following:
(Q1) each Qi,j induces a backwards-transitive path,
s , then v is also an out-neighbour of
(Q2) if v ∈ V (T ) \ (C2 ∪ Vlong ) is an out-neighbour of qi,j
0
s for all s0 ≥ s + 3,
qi,j
10
JAEHOON KIM, DANIELA KÜHN, AND DERYK OSTHUS
0
s , then v is also an in-neighbour of q s
(Q3) if v ∈ V (T ) \ (C2 ∪ Vlong ) is an in-neighbour of qi,j
i,j
for all s0 ≤ s − 3.
|Q
|−1
1 . . . q i,j
For all i ∈ Ilong and all j ∈ [1000k 4 ] we let int(Qi,j ) := qi,j
denote the interior
i,j
1
7
of Qi,j . Let Qi,j , . . . , Qi,j be disjoint segments of int(Qi,j ) such that int(Qi,j ) = Q1i,j . . . Q7i,j ,
3k+3
|Q1i,j | = |Q7i,j | = k + 1, |Q2i,j | = |Q6i,j | = k and |Q3i,j | = |Q5i,j | = k + 2. So qi,j
is the final vertex
|Q
of Q3i,j and qi,j i,j
|−3k−3
is the initial vertex of Q5i,j . We let
Q0i,j := Q1i,j ∪ Q2i,j ∪ Q3i,j ∪ Q5i,j ∪ Q6i,j ∪ Q7i,j
0
and write Vlong
for the set of all those vertices which lie in Q0i,j for some i ∈ Ilong and some
0
j ∈ [1000k 4 ]. Thus Vlong
⊆ Vlong and
0
|Vlong
| ≤ (3k + 3) · 40k · 1000k 4 ≤ 3 · 105 k 6 .
Claim 4.3: There exists an index set IR ⊆ Ilong,α × [1000k 4 ] such that, writing
[
V (Q0i,j ) and IS := (Ilong,α × [1000k 4 ]) \ IR ,
R :=
(i,j)∈IR
for every (i, j) ∈ IS every vertex in Q0i,j has at least k in-neighbours and at least k out-neighbours
in R, and such that |IR | ≤ 700k 3 .
` : i ∈ I
4
To prove Claim 4.3, for each ` ∈ [3k + 3] we consider U ` := {qi,j
long,α , j ∈ [1000k ]}
|Q
|−`
and V ` := {qi,j i,j
: i ∈ Ilong,α , j ∈ [1000k 4 ]}. By Proposition 2.6 applied to T [U ` ], there
`
`
exists a set ZU ⊆ U with |ZU` | ≤ 3k log |U ` | and such that every vertex in U ` \ ZU` has at
least k out-neighbours and k in-neighbours in ZU` . Similarly, there exists a set ZV` ⊆ V ` with
|ZV` | ≤ 3k log |V ` | and such that every vertex in V ` \ ZV` has at least k out-neighbours and k
S
in-neighbours in ZV` . We let Z := `∈[3k+3] (ZU` ∪ZV` ) and write IR for the set of all those indices
(i, j) for which Z contains some vertex in Q0i,j . Let R and IS be as defined in the statement of
Claim 4.3. Then Z ⊆ R and for every (i, j) ∈ IS every vertex in Q0i,j has at least k in-neighbours
and at least k out-neighbours in Z ⊆ R. Moreover,
|IR | ≤ |Z| ≤ (6k + 6) · 3k log(2 · 104 k 5 ) ≤ 700k 3 ,
as required in Claim 4.3.
Let
S :=
[
V (Q0i,j )
[
and B :=
(i,j)∈IS
V (Q0i,j ).
(i,j)∈Ilong,β ×[1000k4 ]
Moreover, let
S 1,7 :=
[
(i,j)∈IS
V (Q1i,j ∪ Q7i,j )
and R1,7 :=
[
V (Q1i,j ∪ Q7i,j ),
(i,j)∈IR
and define B 1,7 similarly. Note that by Claim 4.3 every vertex in S has least k in-neighbours
and at least k out-neighbours in R.
Claim 4.4: We may colour all vertices in S 1,7 ∪ R ∪ B as well as some additional vertices lying
0
outside Vlong
such that
(i) all vertices in S 1,7 are coloured α and all vertices in R ∪ B are coloured β,
(ii) all coloured vertices are safe,
A TOURNAMENT VERSION OF THE LOVÁSZ PATH REMOVAL CONJECTURE
11
(iii) the set C3 consisting of all vertices coloured so far has size |C3 | ≤ 5 · 105 k 6 m and |C3 \
(C2 ∪ S 1,7 ∪ R ∪ B)| ≤ 220k 4 .
To prove Claim 4.4, we first colour all vertices in S 1,7 with α and all vertices in R ∪ B with β.
Recall from (Q1) that {int(Qi,j ) : (i, j) ∈ IR } is a collection of backwards-transitive paths with
|int(Qi,j )| ≥ k + 1. So we may apply the Lemma 2.7 to obtain sets UR and WR such that
(a) UR , WR ⊆ R1,7 ,
(b) |UR |, |WR | ≤ 2k(k + 1),
(c) for any set F ⊆ V (T ) of size at most k − 1, and for every vertex v ∈ V (T ) \ F which lies
on some path in {int(Qi,j ) : (i, j) ∈ IR } there exists a directed path (possibly of length
0) in T [(R1,7 ∪{v})\F ] from v to a vertex in UR and a directed path in T [(R1,7 ∪{v})\F ]
from a vertex in WR to v.
We next apply Lemma 2.7 to the collection of backwards-transitive paths {int(Qi,j ) : (i, j) ∈ IS }
to obtain sets US , WS ⊆ S 1,7 . Finally, we apply Lemma 2.7 to {int(Qi,j ) : (i, j) ∈ Ilong,β ×
[1000k 4 ]} to obtain sets UB , WB ⊆ B 1,7 . Let U := UR ∪ US ∪ UB and define W similarly. Apply
0
0 ∪C )
Claim 2 with C2 , U ∪W , Vlong
playing the roles of C, Z, N to obtain a set Z 0 ⊆ V (T )\(Vlong
2
0
0
and a colouring of the vertices in Z such that every vertex in U ∪ W ∪ Z is safe and
|U ∪ W ∪ Z 0 | ≤ 9k 2 |U ∪ W | ≤ 9k 2 · 12k(k + 1) ≤ 220k 4 .
So the set C3 consisting of all vertices coloured so far satisfies |C3 \ (C2 ∪ S 1,7 ∪ R ∪ B)| ≤ 220k 4
0 | + |C | ≤ 220k 4 + 3 · 105 k 6 + 8000k 4 m ≤ 5 · 105 k 6 m. Using (c) (and its
and |C3 | ≤ 220k 4 + |Vlong
2
analogue for US , WS and UB , WB ) it is now straightforward to check that (ii) holds. (To check
that the vertices in S 1,7 are partition-safe we use that every vertex in S has least k in-neighbours
and at least k out-neighbours in R and that all vertices in R are coloured β.) This completes
the proof of Claim 4.4.
r r
m
Claim 4.5: For each s ∈ [k] there are indices (i`s , js` ), (im
s , js ), (is , js ) ∈ IS such that
S
r
(i) the set s∈[k] {i`s , im
s , is } has size 3k (i.e. all these indices are different from each other),
a
(ii) for each s ∈ [k] and each 2 ≤ a ≤ 6 no vertex in V (Qai` ,j ` ∪ Qaim
m ∪ Qir ,j r ) is coloured,
s ,js
s s
s s
(iii) for each s ∈ [k] there is a directed edge e1s from the initial vertex of Q3i` ,j ` to the initial
s s
2
5
vertex of Q3im
m , and a directed edge es from the final vertex of Qim ,j m to the final vertex
,j
s
s
s
s
of Q5irs ,jsr .
Note that Claim 4.3 implies that for each s ∈ Ilong,α there are at least 1000k 4 − |IR | ≥ 300k 4
indices j ∈ [1000k 4 ] for which (s, j) ∈ IS . Since |C3 \(C2 ∪S 1,7 ∪R∪B)| ≤ 220k 4 by Claim 4.4(iii)
and C2 ∩ Vlong = ∅, we can pick an index j = j(s) with (s, j(s)) ∈ IS and such that the coloured
vertices on int(Qs,j(s) ) are precisely those in Q1s,j(s) ∪ Q7s,j(s) . Let u(s) denote the initial vertex of
|Q
|−2k−2
s,j(s)
2k+2
Q3s,j(s) (so u(s) = qs,j(s)
) and let v(s) denote the final vertex of Q5s,j(s) (so v(s) = qs,j(s)
).
Now consider the subtournament T1 of T which is induced by all the vertices v(s) for all
s ∈ Ilong,α . Thus |T1 | = |Ilong,α | ≥ 18k by Claim 4.2. Together with Proposition 2.1 this implies
that there is a set I1 ⊆ Ilong,α such that |I1 | ≥ 12k and such that for every s ∈ I1 the vertex
v(s) has at least 3k out-neighbours in T1 . We now consider the subtournament T2 of T which
is induced by all the vertices u(s) for all s ∈ I1 . By Proposition 2.1 applied to T2 there is a
set I2 ⊆ I1 such that |I2 | ≥ 6k and such that for every s ∈ I2 the vertex u(s) has at least 3k
in-neighbours in T2 .
12
JAEHOON KIM, DANIELA KÜHN, AND DERYK OSTHUS
m
`
Now let im
1 , . . . , ik be k distinct indices in I2 . For each s ∈ [k] choose an index is ∈ I1
m
m `
`
such that u(i`s ) is an in-neighbour of u(im
s ) and such that the 2k indices i1 , . . . , ik , i1 , . . . , ik
r
r
are distinct. Finally, for each s ∈ [k] choose an index is ∈ Ilong,α such that v(is ) is an outr
r
neighbour of v(im
s ) and such that the indices i1 , . . . , ik are distinct from each other and from
m `
`
im
1 , . . . , ik , i1 , . . . , ik . This completes the proof of Claim 4.5.
We are now ready to prove Claim 4. For each s ∈ [k] let Qs denote the path formed by
7
6
5
4
Q1i` ,j ` ∪ Q2i` ,j ` ∪ Q3im
m ∪ Qim ,j m ∪ Qim ,j m ∪ Qir ,j r ∪ Qir ,j r ,
s s
s
s s
s
s
s
s ,js
s
s
s
s
the initial vertices of both Qi`s ,js` and Q3i` ,j ` , the final vertices of both Qirs ,jsr and Q5irs ,jsr as well as
s
s
the edges e1s and e2s guaranteed by Claim 4.5(iii). Let i0s := i`s and is := irs . Then Qs joins the
sink of Bi0s to the source of Ais , i.e. Claim 4(i) holds.
Recall that all the vertices in Q1i` ,j ` ∪Q7irs ,jsr as well as the two endvertices of Qs are coloured α,
s s
and all other vertices of Qs are uncoloured (i.e. lie outside C3 ). Colour all the (so far uncoloured)
vertices of Qs with α (for all s ∈ [k]) and then all other vertices in Vlong which are still uncoloured
with β. Let C4 be the set of coloured vertices obtained in this way.
= α,
=β
int(Qirs ,jsr )
m)
int(Qim
s ,js
int(Qi`s ,js` )
int(Qi,j )
Q1i,j
Q2i,j
Figure 2 :
Q3i,j
Q4i,j
Q5i,j
Q6i,j
Q7i,j
Colour patterns of the paths int(Qi,j ) with (i, j) ∈ IS in the
case when k = 1. The thick arrows indicate int(Qs ).
Since |C3 \(C2 ∪S 1,7 ∪R∪B)| ≤ 220k 4 by Claim 4.4(iii) and C2 ∩Vlong = ∅, for each s ∈ Ilong,β
there is at least one index j 0 = j 0 (s) such that V (int(Qs,j 0 (s) )) ∩ C3 = V (Q0s,j 0 (s) ). Moreover,
since V (Q0s,j 0 (s) ) ⊆ B, all the vertices in V (Q0s,j 0 (s) ) are coloured β by Claim 4.4(i). Altogether
this shows that all vertices on Qs,j 0 (s) are coloured β. For each s ∈ Ilong,β let Ps := Qs,j 0 (s) .
Together with the short paths Ps for all s ∈ Ishort,β this gives k paths satisfying Claim 4(ii).
Our choice of the paths Qs and Ps implies that Claim 4(iii) holds too.
Let us now check that all vertices in C4 \ C3 are safe. First consider any v ∈ C4 \ C3 which is
coloured α. Then one of the following holds:
(a) v ∈ S \ S 1,7 ,
(b) v ∈ V (Q4im
m ) for some s ∈ [k].
s ,js
Suppose first that (a) holds. So there exists (i, j) ∈ IS such that v ∈ Q2i,j ∪ Q3i,j ∪ Q5i,j ∪ Q6i,j .
Since Qi,j is a backwards-transitive path by (Q1), it follows that every vertex in Q1i,j (except
possibly its final vertex) is an out-neighbour of v and every vertex in Q7i,j (except possibly its
initial vertex) is an in-neighbour of v. Since all vertices in S 1,7 ⊇ Q1i,j ∪ Q7i,j are coloured α and
A TOURNAMENT VERSION OF THE LOVÁSZ PATH REMOVAL CONJECTURE
13
are safe, it follows that v has at least k safe in-neighbours and at least k safe out-neighbours
of colour α. So by (S3) v is forwards- and backwards-safe. Since by Claim 4.3 v has at least k
in-neighbours and k out-neighbours in R (and all vertices in R are coloured β) it follows that v
is partition-safe. So v is safe.
Now suppose that (b) holds. As in (a) one can show that v is forwards- and backwards-safe.
6
Moreover, by (Q1) every vertex in Q2im
m is an out-neighbour of v and every vertex in Qim ,j m is
s ,js
s
s
6
2
an in-neighbour of v. But all the vertices in Qim
m ∪ Qim ,j m are coloured β, so v is partition-safe
,j
s
s
s
s
and thus safe.
Now consider any v ∈ C4 \ C3 which is coloured β. Then one of the following holds:
(c) v ∈ S \ S 1,7 ,
m
(d) v ∈ V (Q4i,j ) for some i ∈ Ilong and j ∈ [1000k 4 ] such that (i, j) ∈
/ {(im
s , js ) : s ∈ [k]}.
If (c) holds then v has at least k in-neighbours and k out-neighbours in R. Since all vertices in
R are coloured β and are safe, this implies that v is safe. Moreover, together with Claim 4.4(ii)
and the safety of the vertices in C4 \ C3 which are coloured α, this implies that all vertices in
0
Vlong
are safe.
m
3
Now suppose that (d) holds. Since (i, j) ∈
/ {(im
s , js ) : s ∈ [k]} all vertices in Qi,j (except
possibly its initial vertex) and all vertices in Q5i,j (except possibly its final vertex) are coloured
0 . By (Q1) every vertex in int(Q3 )
β. Moreover, all these vertices are safe since they lie in Vlong
i,j
is an out-neighbour of v and every vertex in int(Q5i,j ) is an in-neighbour of v. So v is safe. This
completes the proof that all vertices in C4 \ C3 (and thus also all coloured vertices) are safe,
i.e. Claim 4(iv)(α) holds.
Let Cα be the union of V (Q4im
m ) over all s ∈ [k]. Thus the number of vertices of colour α
s ,js
0
outside Cα is at most |C3 | + |Vlong | ≤ 106 k 6 m, i.e. Claim 4(iv)(β) holds. Moreover, if v ∈
6
V (T ) \ C4 and v has an in-neighbour in some V (Q4im
m ) then by (Q3) all vertices in Qim ,j m
s ,js
s
s
6
are also in-neighbours of v. But all vertices in Qim
m are coloured β. So v has at least k
,j
s
s
in-neighbours of colour β. Similarly, if v has an out-neighbour in V (Q4im
m ) then by (Q2) all
s ,js
2
2
vertices in Qim
m are also out-neighbours of v. But all vertices in Qim ,j m are coloured β. So
s ,js
s
s
v has at least k out-neighbours of colour β. This shows that Claim 4(iv)(γ) holds and thus
completes the proof of Claim 4.
The next claim shows that by colouring every uncoloured vertex with β, all vertices will
become safe. Together with Claim 1 this then implies that the partition consisting of the colour
classes Vα , Vβ is as required in Theorem 1.2.
Claim 5: We can colour all uncoloured vertices with β. Then every vertex is safe.
Colour all uncoloured vertices (i.e. all vertices in V (T ) \ C4 ) with β. Consider any vertex
v ∈ V (T ) \ C4 . If v ∈
/ E 0 then by (D3) and (D4) v has an in-neighbour in As and an outneighbour in Bs for every 19k < s ≤ 20k. Since the vertices in all these sets As and Bs are
coloured β and are safe, this implies that v is safe.
0 \E 0 . As above it follows that v has k safe in-neighbours of colour β.
Suppose next that v ∈ EB
A
If v has k out-neighbours of colour β which are lying outside E 0 , then these out-neighbours are
safe and so v is safe. So suppose that v has less than k out-neighbours of colour β which are lying
outside E 0 . Recall from Claim 4(iv)(β) that at most 106 k 6 m vertices of colour α lie outside the
set Cα . Together with the fact that δ̂ + (T )−|E 0 | ≥ 5·106 k 6 m ≥ k +106 k 6 m by (3.4), this implies
14
JAEHOON KIM, DANIELA KÜHN, AND DERYK OSTHUS
that v has an out-neighbour in Cα . But now Claim 4(iv)(γ) implies that v has k out-neighbours
of colour β in C4 . Since all the vertices in C4 are safe, this shows that v is safe.
0 . As in the previous case one can show that v has k safe outFinally, suppose that v ∈ EA
0 , then these
neighbours of colour β. If v has k in-neighbours of colour β which are lying outside EA
in-neighbours are safe and so v is safe. So suppose that v has less than k in-neighbours of colour β
0 . Together with the fact that δ̂ − (T )−|E 0 | ≥ 5·106 k 6 m ≥ k +106 k 6 m
which are lying outside EA
A
by (3.3), this implies that v has an in-neighbour in Cα . Thus Claim 4(iv)(γ) implies that v has
k in-neighbours of colour β in C4 . Since all the vertices in C4 are safe, this shows that v is safe.
This completes the proof of Claim 5 and thus of Theorem 1.2.
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Jaehoon Kim, Daniela Kühn, Deryk Osthus
School of Mathematics
University of Birmingham
Edgbaston
Birmingham
B15 2TT
UK
E-mail addresses: {kimJS, d.kuhn, d.osthus}@bham.ac.uk